Active carbody roll control in railway vehicles using
hydraulic actuation
Edoardo F. Colombo, Egidio Di Gialleonardo, Alan Facchinetti, Stefano Bruni
nDipartimento di Meccanica, Politecnico di Milano, Via La Masa, 1 I-20156 Milan, Italy
Received 1 October 2013 Accepted 30 May 2014
1. Introduction
Recent decades have seen increasing integration of electronics and control into rail vehicles which used to be purely mechanical
systems (Goodall, 2011). As part of this process, mechatronic
technologies are increasingly being introduced in the running gear to improve running dynamics, service performance and ride
quality (Bruni, Goodall, Mei, & Tsunashima, 2007; Goodall, Bruni,
& Facchinetti, 2012).
In this regard, particularly noteworthy is the development of concepts for active secondary suspension (i.e. the suspension stage
that isolates the carbody from the bogies (Goodall et al., 2012)),
with the aim of raising service speeds on existing networks with the same level of ride comfort. Tilting carbody technology is an obvious example of this trend, but requires a rather complex bogie
and suspension design based on a tilting bolster (Persson, Goodall,
& Sasaki, 2009), which leads to increased vehicle weight and, given the constraints applied to the vehicle gauge, reduces the space available for passengers.
For these reasons, tilting body technology in the strict sense is not used for very high-speed trains, where severe constraints
apply in terms of maximum axle load. For this specific application
simpler and lighter carbody roll actuation systems have been proposed, aiming at permitting limited tilt angles (in the range
of 1–21, compared with the 6–81 maximum tilt angle achieved by
tilting trains in the strict sense) that are nevertheless sufficient to
increase service speed significantly on existing lines.
An example of this concept is the Japanese high-speed train
Shinkansen Series N700, which attains approximately 11 tilting by
actively controlling the pneumatic secondary suspension (Nakakura
& Hayakawa, 2009; Tanifuji, Koizumi, & Shimamune, 2002).
The same concept was proposed inAlfi, Bruni, Diana, Facchinetti,
and Mazzola (2011), also showing the possible use of active roll actuation to reduce overturning risk in the presence of crosswind. Pneumatic actuation via secondary air springs is, however, not free
from drawbacks:firstly, the actuation pass-band is limited, which
may lead to delays in tracking the desired tilt reference. Secondly, the coupling of vertical and roll carbody motion and the dynamics of the pneumatic system cause disturbances affecting ride comfort
and potentially leading to instability problems (Facchinetti,
Di Gialleonardo, Resta, Bruni, & Brundisch, 2011; Tanifuji, Saito, Soma, Ishii, & Kajitani, 2009). Thirdly, with this actuation concept the use of a mechanical anti-roll bar cannot be avoided and the restoring roll torque generated by this component counteracts roll actuation, resulting in smaller tilt angles and increased energy consumption (Facchinetti et al., 2011).
This paper proposes an active anti-roll device to replace the passive anti-roll bar, making it possible actively to tilt the carbody. nCorresponding author.
The use of an active anti-roll bar in rail vehicles wasfirst suggested byPearson, Goodall, and Pratt (1998), with a modified layout of a mechanical bar in which either linear actuators were introduced to replace the links to the carbody, or a rotary actuator was placed in series with the torsion bar. This original concept, however, is
difficult to make fault tolerant, which is a major drawback. For this
reason, in this paper hydraulic actuation is proposed instead, using cross-connected actuators so that a roll torque can be generated for a theoretically null vertical force: in this way it is easier to reject track irregularity related disturbances and, at the same time, tilt actuation becomes faster, more accurate and less energy consuming compared with pneumatic actuation. The hydraulic system can be dimensioned to provide the same roll stiffness as a conventional anti-roll bar when operated in the passive mode (Colombo, Di Gialleonardo, Facchinetti, & Bruni, 2013).
Three different control strategies are proposed for the active
hydraulic anti-roll device and for all three control gains are defined
based on Genetic Algorithm (GA) optimisation, having as multiple objectives the tracking of a reference carbody tilt angle, optimising ride comfort and keeping the energy required for actuation within
acceptable limits. The reference tilt angle is defined on the basis of
vehicle speed and curve geometry (curvature, cant, length of transitions), under the assumption that this information is made available to the control unit through geo-localisation of the train and mapping of the line, as implemented in the Shinkansen N700
train (Nakakura & Hayakawa, 2009). Other ways of defining the
reference tilt angle, i.e. based on inertial sensors, would be possible and are in use, but it is not the purpose of this paper to investigate
or discuss the benefits and drawbacks of alternative methods
available to accomplish this task.
2. Concept of the active hydraulic anti-roll device
The active anti-roll device is designed to accomplish two main functions:
actuate the desired carbody tilt angle when the vehiclenegoti-ates a curve;
provide the same carbody to bogie roll stiffness as aconven-tional anti-roll bar device when carbody tilt is not required.
A scheme of the device is shown inFig. 1for one bogie and
includes the following main components:
two linear hydraulic actuators AL and AR, placed at the left andright side of the bogie and connecting the bogie frame with the carbody;
a main hydraulic circuit (thick solid lines in thefigure),cross-connecting the chambers of the actuators. Two reservoirs R1 and R2 are introduced in the circuit, and their volume is designed to provide the desired degree of roll compliance in passive mode;
a hydraulic feeder circuit (thin solid line in the figure) tocontrol fluid flow in the reservoirs with a pump (PP) and a
servo-valve SV.
The working principle in passive mode is the same as for hydraulically interconnected suspensions developed mainly for
automotive applications (Zhang, Smith, & Jeyakumaran, 2010). The
upper chamber of the left actuator (LU) is connected to the lower chamber of the right actuator (RL) and the lower chamber of the left actuator (LL) with the upper chamber of the right actuator (RU). In this way, the hydraulic suspension nominally provides zero stiffness in the vertical direction and a non-zero roll stiffness. A slow upward bouncing movement of the carbody will reduce the oil volume in the two upper chambers LU and RU and will increase the volume in the two lower chambers LL and RL by the same amount. Thanks to the cross-connection of the chambers, oil
willflow from LU to RL and from RU to LL and, neglecting viscosity
effects, no variation of the pressure in the chambers will occur. In a
slow downward bouncing motion of the carbody, oil will flow
from LL to RU and from RL to LU and again no pressure variation and hence no vertical force will be generated.
When instead the carbody performs a roll rotation with respect to the bogie for example counter-clockwise, the volumes in chambers LL and RU will decrease while those in LU and RL will increase. As a consequence, oil pressure will increase in LL and RU and decrease in LU and RL, producing a restoring moment in the roll direction.
The active roll function is only activated during curve negotia-tion and is implemented by the actuated servo-valve SV which controls the volume of oil in the two branches of the main hydraulic circuit, extending the linear actuator on one side while contracting the other actuator, thus providing carbody tilt. The
reference carbody tilt angle is defined based on the cant deficiency
of the curve which, in turn, depends on the curve geometry and on the vehicle speed. The position of the vehicle along the curve is assumed to be known from a positioning system, in the same way
as inNakakura and Hayakawa (2009).
The active hydraulic anti-roll device is used in conjunction with an active hydraulic secondary lateral suspension, whose function is to reduce the unloading of the inner wheels caused by the lateral displacement of the carbody and to prevent the lateral bump-stops to come in contact with the carbody, which would degrade ride quality. The active lateral suspension envisaged here is a simple“hold-off” type (Bruni et al., 2007), generating a lateral
force in open loop which is defined to be proportional to the cant
deficiency. Because of the simple control strategy envisaged for the
active lateral suspension, the integration of active tilt and active
lateral control at the secondary suspension as proposed inZhou,
Zolotas, and Goodall (2011) is not pursued here.
3. Mathematical model
A detailed non-linear model of the active hydraulic anti-roll device was derived and interfaced with a non-linear mechanical model of the entire rail vehicle, to optimise the controller para-meters using the Genetic Algorithm and to assess numerically the performance of the actuated vehicle.
SV RU LU RL LL R2 R1 AL AR PP to bogie to carbody to bogie to carbody
3.1. Model of the active hydraulic anti-roll bar
The following assumptions were made in developing a math-ematical model of the hydraulic anti-roll bar:
(i) laminar and mono-dimensionalflow in all pipes, losses due
tofluid viscosity are neglected;
(ii) isothermal conditions of thefluid;
(iii) partially compressiblefluid in the actuator chambers;
(iv) incompressiblefluid in the pipes and reservoir;
(v) constant values for the supply and return pressures; (vi) negligible inertial effects in the piston and piston rod; (vii) servo-valve assumed to behave as a 1st order system.
Based on assumptions (i)–(vii), a 3rd order model is derived for
the active suspension. Two state variables are introduced for
the fluid pressure p1 in reservoir R1 and chambers RU and LL,
and thefluid pressure p2in reservoir R2 and chambers LU and RL.
The third state variable, xs, is the position of the spool in the
servo-valve.
Considering conservation of mass for thefluid volume contained
in each branch of the main hydraulic circuit and considering
assumptions (i)–(iv), the following state equations are obtained
(details are provided inAppendix A):
2V0 β þ AðurulÞ β _p1þ2ðciþceÞp12cip2¼ Að_ur _ulÞþQs1 ð1Þ 2V0 β AðurulÞ β _p22cip1þ2ðciþceÞp2¼ þAð_ur _ulÞþQs2 ð2Þ
where V0is the totalfluid volume contained in each branch of the
main circuit when the suspension is in the reference position,β is
the bulk modulus, A is the area of the piston, uland urare the
displacements of the piston in the left and right actuators, ciand ce
are the internal and external leakage coefficient of the piston
(Merritt, 1967) and, finally, Qs1 and Qs2 are the volumetric flow
rates corresponding to an inlet/outlet of fluid produced by the
servo-valve in the two branches of the circuit (seeFig. A1). Theflow
rate quantities Qs1 and Qs2 are non-linearly dependent on the
pressure drop across the servo-valve and on the position xsof the
servo-valve spool: Qs1;2¼ CsðxsÞdsxs ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ðpS p1;2Þ ρ q
for fluid supply CsðxsÞdsxs
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2ðp1;2 pRÞ
ρ q
for fluid return 8
> < >
: ð3Þ
with pSand pRthe supply and return pressure in the feeding circuit,
Csthe efflux coefficient, which varies with the position of the spool,
and dsa constant length parameter characteristic of the servo-valve.
The servo-valve dynamics is described by the 1st order equa-tion: _xsþ 1 τs xs¼ ks τs us ð4Þ
with usthe command to the servo-valve,τsthe time constant of
the servo-valve and ksa characteristic gain of the servovalve.
Replacing Eq.(3)in(1) and (2)and adding Eq.(4)a set of
non-linear 3rd-order state equations is obtained, which describes the pressure variations in the two branches of the circuit as functions
of the actuators' displacements uland ur(applied by the vehicle,
seeSection 3.2) and of command us(defined by the regulator, see
Section 4).
The forces generated on the left and right pistons arefinally
defined as:
Fl¼ Aðp1p2Þ
Fr¼ Aðp1p2Þ ð5Þ
3.2. Model of the actuated vehicle
The active suspension model described in Section 3.1was
inter-faced with a multi-body model of a complete rail vehicle which was
defined using the in-house developed simulation software ADTreS
(Bruni, Collina, Diana, & Vanolo, 1999; Di Gialleonardo, Braghin, & Bruni, 2012). The rail vehicle model consists of one carbody, two bogies and four wheelsets, all considered as rigid bodies, and of primary and secondary vehicle suspensions, modelled as linear and non-linear lumped parameter elastic and viscous elements.
Each rigid body is assigned five degrees of freedom (vertical
and lateral displacements, yaw, roll and pitch rotations), while the forward motion of the body's centre of mass is assumed to have constant speed V. Therefore, the model has a total of 35 degrees of freedom.
Wheel–rail contact forces are introduced according to a
non-linear multi-Hertzian contact model (Braghin, Bruni, & Diana,
2006). Excitation introduced by track irregularity is introduced
in the simulation by considering a time history of vertical and lateral displacements applied to the contact points on the rails,
according to a random irregularity profile generated based on the
ERRI spectra for track irregularities considering the low level
irregularity case (ERRI: B176/3, 1993). Trackflexibility effects are
not considered in the simulation, as these would mainly affect the high-frequency component of the vehicle dynamics (above 20 Hz) (Di Gialleonardo et al., 2012), whereas the interest here lies in the
frequency range up to 5–10 Hz.
The motion of each body is described with respect to a moving reference travelling with constant speed along the track centre-line. The bodies are assumed to undergo small displacements relative to the moving reference so that the equations of motion for the vehicle can be linearised with respect to kinematical non-linear effects only, and take the symbolic form:
Mv€zvþCv_zvþKvzv¼ FiðV; tÞþFnlðzv; _zvÞþFcðzv; _zv; V; tÞ ð6Þ
where zv is the vector of the vehicle's 35 coordinates shown in
Fig. 2(yaw rotations are not depicted for the sake of clarity), Mv, Cv
and Kvare the mass, damping and stiffness matrices of the vehicle
respectively, Fi is the vector of inertial forces due to the
non-inertial motion of the moving references (this term reduces to zero when the vehicle is considered to move along a tangent track), V is
the vehicle speed, Fnl is the vector of forces due to non-linear
elements in the suspensions (including the active suspension
considered here) and Fc is the vector of the generalised forces
due to wheel–rail contact. More details on the derivation of the
equations of motion for the vehicle can be found inBraghin et al.
(2006),Bruni et al. (1999), andDi Gialleonardo et al. (2012). A co-simulation is established between the vehicle model
described by Eq.(6)and the models of the two active suspensions
on the leading and trailing bogie. To this end, the piston displace-ments uð Þll, uð Þrl, u t ð Þ l and u t ð Þ
r (with superscripts l and t referring to the
leading and trailing bogie respectively) are derived from the vehicle coordinates according to the following equation, see Fig. 2 for the nomenclature of geometric parameters used and for sign conventions:
uðlÞl ¼ xðcÞþlϑðcÞ y bϑðcÞz xðlÞþbϑðlÞz uðtÞl ¼ xðcÞlϑðcÞ y bϑðcÞz xðtÞþbϑðtÞz uðlÞr ¼ xðcÞþlϑðcÞ y þbϑðcÞz xðlÞbϑðlÞz uðtÞr ¼ xðcÞlϑðcÞy þbϑðcÞz xðlÞbϑðlÞz ð7Þ
The time derivatives of the piston displacements are obtained
by straightforward derivation with respect to time of Eq.(7).
The forces generated by the actuators of the hydraulic suspen-sion give rise to non-zero resultant vertical forces on the carbody,
leading and trailing bogie, denoted respectively as FðcÞx , FðlÞx and FðtÞx , which are expressed as:
FðcÞx ¼ ðFðlÞl þFðlÞr þFðtÞl þFðtÞr Þ FðlÞx ¼ FðlÞl þFðlÞr
FðtÞx ¼ FðtÞl þFðtÞr ð8Þ
where FðlÞl;r are the forces generated on the left and right side of the leading bogie and FðtÞl;r the forces generated by the suspension on the trailing bogie. Furthermore, the actuators' forces give rise to resultant roll moments on the carbody and bogies MðcÞz , MðlÞz and MðtÞz and to a
pitch moment on the carbody MðcÞy , whose expressions are:
MðcÞz ¼ FðlÞ l F ðlÞ r þFðtÞl F ðtÞ r b MðcÞy ¼ FðlÞl FðlÞr þFðtÞl þFðtÞr l MðlÞz ¼ FðlÞl FðlÞr b MðtÞz ¼ FðtÞl FðtÞr b ð9Þ
The forces and moments defined by Eqs.(8) and (9)are applied
to the carbody and bogies and appear in vector Fnl of Eq.(6).
The state vector x and input vector u for the model of the
actuated vehicle are defined as follows:
x¼ zT v _z T v pðlÞ1 p ðlÞ 2 x ðlÞ s pðtÞ1 p ðtÞ 2 x ðtÞ s n oT ð10Þ u¼ uðlÞ s uðtÞs n oT ð11Þ with pðlÞ1, pðlÞ2, xðlÞs and pðtÞ1, pðtÞ2, xðtÞs the states of the active anti-roll bar on the leading and trailing bogie respectively, and uðlÞs, uðtÞs the servo-valve commands of the active suspension in the two bogies.
Since zvis made up of 35 coordinates, the total number of states in
the model is 76.
Eqs. (6)–(9) together with Eqs. (1)–(4) for the leading and
trailing bogie can be rewritten in standard state form:
_x ¼ fðxÞþgðxÞu ð12Þ
Eq.(12)is integrated numerically in the time domain, leading to
the simulation of the nonlinear dynamic behaviour of the complete actuated vehicle. The servo-valve commands uðlÞs and uðtÞs are defined by the regulator according to the control strategies described in the next section.
4. Control strategies
The main objective of the control system is to track a reference carbody tilt angleϑðcÞx;ref, defined on the basis of the geometry of the curve being negotiated and of the vehicle speed. Assuming the vehicle to run at constant speed, the reference tilt angle results in
a trapezoidal profile, with ramps occurring during the negotiation
of the entry and exit transition spirals and a constant value in full curve (seeFig. 3).
Three alternative control strategies are considered: – feed-forwardþPID feed-back (PID);
– feed-forwardþPID feed-backþSky-Hook damping (PIDSH); – Linear Quadratic controller with integral feedback (LQ).
4.1. Feed-forwardþPID feed-back regulator
This is the simplest control strategy considered: the command
to the servo-valve us is defined as the sum of a feed-forward
contribution uFFand of a PID feed-back term uFB(seeFig. 4). The
feed-forward contribution, representing the main contribution to
Fig. 3. Reference tilt angle and its time derivative along the curve. l x(c) θy(c) x(t) θy(t) x(l) θy(l) b x(c) y(c) θz(c) x(l,t) y(l,t) θz(l,t) x(1) θy(1) x(2) θy(2) x(3) θy(3) x(4) θy(4) x(1,2,3,4) y(1,2,3,4) θz(1,2,3,4)
the control action, aims to provide a fast response of the actuated system, whereas the feed-back contribution aims to compensate
the effect of the simplifying assumptions used to define the
feed-forward component and the effect of the external disturbances mainly caused by track irregularity.
Infirst approximation, i.e. by considering the fluid as
comple-tely incompressible and neglecting piston leakages, the hydraulic system behaves as an integrator. For this reason, the feed-forward
contribution uFFis defined as proportional to the time derivative of
the reference carbody tiltϑx,refand takes a constant non-zero value
in the transition spirals and zero value in full curve owing to the shape of the reference tilt profile (seeFig. 3).
For the feed-back contribution uFBa PID regulator is adopted,
acting on the error between the reference tilt angle and the relative carbody-bogie rollΔϑx:
uFB¼ kP ϑðcÞx;refΔϑx þkD _ϑðcÞx;refΔ_ϑx þkI Z ϑðcÞ x;refΔϑx dt ð13Þ with
Δϑx¼ ϑðcÞx ϑðlÞx for the leading bogie
Δϑx¼ ϑðcÞx ϑðtÞx for the trailing bogie ð14Þ
The regulator gains kP, kDand kIin(13)are defined by means of
Genetic Algorithm based optimisation, as described inSection 5.1.
Note that, in order to use a practical measure, the error in(13)is
defined based on the carbody roll rotation relative to the bogie
instead of relative to the top-of-rail plane. As a consequence, the
feedback command uFB is different for the leading and trailing
bogies. For the same reason, a positioning error due to the roll
rotation of the bogies is expected; this error is however small owing to the considerable stiffness of the primary suspension.
4.2. Feed-forwardþPID feed-back regulatorþSky-Hook damping
Tilting capabilities should be introduced without increasing, or even reducing, the level of carbody vibration in response to the transient excitation induced by curve negotiation and to track irregularities. For this reason, a second control strategy is
con-sidered, which is the same as the one described inSection 4.1with
the addition of Sky-Hook damping (seeFig. 5).
Thanks to the pseudo-integrator introduced by the hydraulic
system, the sky-hook damping contribution can be defined as
proportional to the carbody absolute roll acceleration €ϑðcÞx : uFB¼ kP ϑðcÞx;refΔϑx þkD _ϑðcÞx;refΔ_ϑx þkI Z ϑðcÞ x;refΔϑx dtkSH€ϑðcÞx ð15Þ
thus requiring the additional measure of €ϑðcÞx besides the
carbody-roll relative rotationsΔϑxmeasured on the two bogies. The gain kSH
associated with the sky-hook damping contribution is added to the
parameters to be defined through GA optimisation (Section 5.1).
4.3. Linear Quadratic regulator with integral feedback (LQ) The third and last control strategy considered consists of a Linear Quadratic regulator with integral feed-back action (seeFig. 6).
Fig. 4. Block diagram of the feed-forwardþPID feed-back regulator.
Fig. 5. Block diagram of the feed-forwardþPID regulator with Sky-Hook damping (PIDSH).
The integral gain kIand the gain matrix K are computed by
solving an infinite horizon LQ problem, that consists of minimising
the following cost function: ~J ¼1 2 Z1 0 ð ~x T Q~x þuTsRusÞdt ð16Þ
where the augmented state vector ~x ¼ ½ xhc zT is composed of
the state vector xhcof the hydraulic system and of the carbody,
neglecting the bogie dynamics, and of the integrals of the error z¼RðϑðcÞ
x;refΔϑxÞdt.
The weight matrices Q and R are defined as diagonal matrices.
In this case, the diagonal terms of these matrices are the para-meters chosen by means of GA optimisation. The implementation of LQ requires measurement of carbody motion, relative to the bogie, and of the pressures inside the hydraulic circuit.
5. Simulation results
In this section, the mathematical model of the actuated vehicle
defined inSection 3is used to define the gains of the regulators
introduced inSection 4by means of Genetic Algorithm
optimisa-tion. Then, the performance of the actuated vehicle is assessed by numerical simulation. The vehicle considered for this assessment
is afictitious but realistic high-speed vehicle.Table 1reports the
parameters of the active hydraulic suspension and some main parameters of the vehicle considered. Both regulator gain optimi-sation and performance assessment are carried out considering one single high-speed curve geometry with radius 5500 m and cant 105 mm, which is very common in the Italian high-speed network and represents for this scenario the most demanding case for testing the performance of active suspension.
5.1. Optimisation of regulator gains
Optimisation was used to find the most suitable control
parameters. Since the solution space is large, reasonably full of local optima and non-linear, a global optimisation method was preferred to local ones. Genetic algorithms have proven to be a
suitable tool for regulator optimisation in general (Fleming &
Purshouse, 2002) and specifically for active suspensions in ground
vehicles (Baumal, McPhee, & Calamai, 1998; He & McPhee, 2005;
Tsao & Chen, 2001).
Genetic algorithms can be used to optimize either a single fitness function or multiple ones: in the first case they are called single objective genetic algorithms (SGA), in the second multi-objective genetic algorithms (MOGA). The second option is adopted
here, leading to the definition of Pareto front solutions among
which the most appropriate set of regulator gains are identified.
Therefore, fourfitness functions were chosen for the optimisation:
1. maximum absolute error in trajectory tracking;
2. PCT index for seated passengers:, this is a measure of ride
comfort on curve transitions, adopted especially for tilting
trains, and is defined as a non-linear function of the maximum
lateral acceleration, the maximum lateral jerk and the max-imum carbody roll speed perceived by the passengers during
the negotiation of transition curves (CEN EN 12299, 2009), with
higher PCTvalues being associated with a greater probability of
passengers suffering motion sickness;
3. r.m.s. of the lateral vibration perceived by seated passengers; 4. r.m.s. of actuation command given to servo-valve.
Thefirst objective defines the reference signal to be followed
accurately during curve negotiation and the reduction of the quasi-static lateral acceleration in curve to be effectively achieved. The second and third objectives are meant to ensure appropriate ride comfort level for passengers, with reference to the negotiation of curve transition and to the response to track irregularity respectively. The last objective ensures that energy consumption entailed by active carbody tilt is kept within acceptable limits. It is
worth pointing out that, while all of the first three objectives
relate to ride comfort for passengers, they cover different aspects
that can be conflicting, thus requiring the definition of an
appro-priate trade-off when setting tilt control, as also pointed out in Zolotas, Goodall, and Halikias (2007). This is why these objectives are considered separately in the optimisation.
The variables to which the multi-objective optimisation is applied are the regulator gains, i.e. the proportional, integral and derivative gains for the PID regulator, the same three parameters plus a skyhook damping gain for the PIDSH strategy and the six diagonal elements in the weight matrix Q of the LQ regulator.
The algorithm chosen for the optimization of the regulators is
the non-sorting genetic algorithm II (NSGA-II) (Deb, Agrawal,
Pratap, & Meyarivan, 2000), one of the best-performing MOGA approaches in terms of convergence speed and elitism (ability of preserving the best individuals).
Each generation of the genetic algorithm process is ranked
against all fourfitness functions, and a multi-dimensional Pareto
frontier is defined as approximated by the discrete population
formed by the non-dominated individuals in the considered generation.
The genetic algorithm operators were chosen empirically by trial and error. The population size was 150 individuals for all the regulators. Selection was tournament-based, two individuals at a time, without replacement; crossover was single point and poly-nomial mutation had a 10% probability. The optimisation was terminated at 150 generations, which assured convergence to a stable Pareto front approximation.
The NSGA-II algorithm produced a set of optimal solutions
belonging to a 4-dimension Pareto surface.Figs. 7–9show some of
the projections in 2-dimension of the solutions found, for the PID,
PIDSH and LQ regulators respectively. For the PID regulator (Fig. 7),
significant positive correlation is observed for the tracking error
and the PCTindex, so that a decrease of the tracking error generally
leads also to an improved level of ride comfort. The r.m.s. lateral acceleration (not displayed for the sake of brevity) shows limited
variations across the set of identified optimal solutions. Moreover,
the r.m.s. of the actuation command is increasing for decreasing
PCTand tracking error, showing as expected that more actuation
energy is required to improve the performance of the active
suspension. In the central and right plots of Fig. 7a “knee” is
observed corresponding to a PCTvalue of 2.4, a tracking error of
7.7 mrad and an r.m.s. of actuation of approximately 0.028 V. On this point, further improvements of the PCT and tracking error can be obtained only by applying a much larger actuation.
An examination of the set of optimal solutions found for the
PIDSH regulator (cf.Fig. 8), shows it is possible to achieve lower
PCTvalues (and hence improved ride comfort levels) compared to
the PID regulator, at the price of an increase of the actuation
Table 1
Parameters of the active suspension and main parameters of the rail vehicle.
Variable name Symbol Value
Fluid volume in each main branch V0 1.5 102m3
Area of the piston A 1.963 103m2
Oil bulk modulus β 1.1 109
Pa
Supply pressure pS 198 105Pa
Return pressure pR 2 105Pa
Distance between actuators b 2.57 m
Carbody mass 40,700 kg
Carbody roll moment of inertia 8.8 105 kg m2
command r.m.s. In this case, the trend of the PCT index with the tracking error is not monotone and, compared to the PID regulator,
larger tracking errors need to be incurred to achieve PCTvalues in
the range of 1.5 or lower.
Finally, for the LQ regulator (cf.Fig. 9), slightly higher PCTand
tracking error values are obtained compared with the other two
regulators. As will be further discussed in Section 5.2, this is
mainly because the LQ regulator does not include the feed-forward
contribution uFF, and hence performance in terms of reference
tracking and ride comfort is affected by a longer delay in controller action during the negotiation of the entry transition spiral.
In conclusion,five configurations from the GA optimisation of the
three controllers are retained for further analysis in Section 5.2.
To this end, two levels of r.m.s. of actuation are chosen, the lower one
corresponding to the“knee” observed inFig. 7and the second one
approximately three times greater. Then, for each regulator, the
optimal gain set providing the minimum PCTvalue for the chosen
level of actuation is selected. For the LQ regulator, no solution is available for the lower level of actuation, and hence only one
configuration is considered, corresponding to the higher r.m.s. of
actuation. The configurations selected for further analysis are shown
inFigs. 7–9by square markers (low actuation level) and triangular markers (high actuation level).
5.2. Analysis of the optimised controllers
To evaluate and compare the effect of the three control
techniques, a pole plot is shown in Fig. 10 for the optimised
0.5 1 1.5 2 2.5 3 3.5 0 5 10 15 PCT [−]
tracking error [mrad]
0.5 1 1.5 2 2.5 3 3.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 PCT [−] r.m.s. of actuation [V] 0 5 10 15 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
tracking error [mrad]
r.m.s. of actuation [V]
Fig. 7. 2-D projections of the optimal surface for the PID controller.
0.5 1 1.5 2 2.5 3 3.5 0 5 10 15 PCT [−]
tracking error [mrad]
0.5 1 1.5 2 2.5 3 3.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 PCT [−] r.m.s. of actuation [V] 0 5 10 15 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
tracking error [mrad]
r.m.s. of actuation [V]
Fig. 8. 2-D projections of the optimal surface for the PIDSH controller.
0.5 1 1.5 2 2.5 3 3.5 0 5 10 15 PCT [−]
tracking error [mrad]
0.5 1 1.5 2 2.5 3 3.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 PCT [−] r.m.s. of actuation [V] 0 5 10 15 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
tracking error [mrad]
r.m.s. of actuation [V]
controllers. Conclusions can also be drawn on the stability proper-ties of the optimised controllers. This point is especially relevant for the PID with sky-hook controller, since a sky-hook controlled suspension can become unstable in real active isolation systems (Elliott, Serrand, & Gardonio, 2001). The pole plot is shown for a 4 degree-of-freedom half-vehicle model considering the lateral dis-placement and roll rotation of the half car body and of one bogie. For clarity, only the poles representing the low-frequency beha-viour of the vehicle are shown.
All the optimised controllers were considered in the analysis,
but for the sake of brevityFig. 10shows only those with a higher
command r.m.s. Crosses represent the dynamics of the vehicle actuated in open loop: a real pole can be seen close to the origin of the diagram and is associated with the dynamic of the hydraulic anti-roll bar. Two complex conjugate pole couples describe the first lateral and first roll mode of the suspension. The two poles situated approximately at 5 rad/s are associated with the lateral vibration of the carbody, while the other two at 9 rad/s are related to the roll motion of the carbody.
The poles of the vehicle actuated in closed loop show an
additional pole due to the dynamics of the low-passfilter. The
poles of the optimised solution with the PID controller, shown by circles, are close to those of the vehicle actuated in open loop as far as the lateral vibration of the car body is concerned, but show a
significantly lower real part and hence lower damping for car body
roll. The poles of the PID with sky-hook damping optimised controller are shown as triangles. The real and imaginary part of the poles associated with lateral car body motion are again close to those of the vehicle actuated in open loop, while the poles associated with car body roll have lower real and imaginary parts, showing mainly a reduction of the natural frequency for approxi-mately the same level of damping.
Finally, the poles of the vehicle with LQ controller (squares) show an increased damping of the lateral car body motion while preserving car body roll dynamics similar to those of the vehicle actuated in open loop.
Some conclusions can be drawn from the analysis of the pole diagram: all the optimised controllers provide acceptable stability margins, the lateral vibration of the car body is only weakly affected by the type of controller while larger differences are found in terms of roll car body motion. Finally, the stability of the PID with sky-hook controller is not seriously compromised by the flexibility of the primary suspension, as could have been expected
from earlier literature (Elliott et al., 2001). This is because on
railway vehicles the primary suspension is much stiffer than the secondary suspension, so the effect of its compliance on stability is less.
5.3. Performance evaluation for the optimised regulators
This section reports the results of time domain simulations for
selected regulator configurations considering the active vehicle
run-ning at 340 km/h along a curve having the same geometry already
considered inSection 5.1. For comparison, the behaviour of a
conven-tional passive vehicle running at 300 km/h is also considered. Fig. 11shows the roll angle of the carbody relative to the top-of-rail plane as a function of the vehicle position along the track,
considering four different vehicle configurations: the passive
vehicle (top left), and the active vehicle considering the three
control strategies introduced inSection 4: PID (top right), PIDSH
(bottom left) and LQ (bottom right). For all active vehicle con
fig-urations, results are shown for the gain configuration
correspond-ing to the high level of r.m.s. of actuation.
As concerns the dynamics of the passive vehicle, it can be seen that when the vehicle enters the curve the carbody begins to rotate outwards (positive values of the roll angle) owing to the inertial force. Considering instead the active vehicle, the carbody is actuated to follow the reference tilt angle, shown as a dotted line, directed towards the inside of the curve. It can be seen that all controllers are able to track the reference with good accuracy, although the LQ controller shows a delay which is due to the lack
of the feed-forward contribution. As anticipated in Section 4, all
regulators show a small steady state error in full curve because the
error on carbody rotation is defined on the basis of the relative and
not the absolute roll angle.
The lateral acceleration felt by the passengers is shown in Fig. 12for the same vehicle configurations previously described. It is apparent that carbody tilt is able to adequately compensate for
the larger cant deficiency caused in the active vehicle by the
increase of speed. Furthermore, the PIDSH and LQ controllers are also able to compensate for the increased level of high-frequency excitation produced by track irregularity, which increases with the
square of speed. The simpler PID controller is less efficient in
reducing the high-frequency dynamicfluctuations of the
accelera-tion signal, resulting in slightly worse ride comfort levels com-pared with the other two control strategies, as further discussed below.
The ride comfort indices (PCTand r.m.s. of the lateral acceleration)
and the average power (per bogie) required for actuating the carbody
are compared inTable 2for the vehicle with active suspension running
at 340 km/h and for the passive vehicle running at 300 km/h. Active tilt control improves the PCTdespite the increase of speed, while r.m.s.
lateral acceleration is only slightly increased. It can be concluded that the active suspension allows the curve to be negotiated 40 km/h faster with comparable ride comfort levels.
As far as the comparison of alternative regulator configurations
is concerned, for the PID regulator only a marginal improvement of
the PCT index is obtained if the gain settings corresponding to a
higher level of actuation are used, at the expense of more power being required for actuation. On the other hand, for the PIDSH
regulator performance can be improved significantly in terms of
both comfort indices by increasing actuation intensity. The LQ regulator is effective in reducing the components of carbody vibration generated by track irregularities and hence r.m.s. lateral acceleration, but is less effective than the other two regulators in
terms of PCT value, owing to the delay in actuation mentioned
above. Overall, the PIDSH regulator with high actuation level produces the best ride comfort performance, with an acceptable level of actuation power.
Simulations also showed that the use of active tilt (in conjunction with an active lateral secondary suspension) does not cause running safety issues: the track shift forces and the derailment and
over-turning coefficients were verified to remain well within the
admis-sible ranges, despite the increase of speed from 300 to 340 km/h.
−5 −4 −3 −2 −1 0 1 −15 −10 −5 0 5 10 15 α [1/s] ω [rad/s] Open−loop PID PID+SH LQ
6. Conclusions
This paper proposed a novel concept for an active hydraulic anti-roll bar for use in high speed railway vehicles, with the aim of applying a limited amount of carbody tilt and reducing the exposure of passengers to non-compensated lateral acceleration in
curves. Three control strategies were defined for the active
suspen-sion, and their parameters were chosen based on multi-objective Genetic Algorithm optimisation using a multi-body model of the actuated vehicle. Four objective functions were considered in the optimisation process, representing quantitative measurements of the tracking error, ride comfort and energy required for actuation.
0 500 1000 1500 2000 −0.5 0 0.5 1 [m] [m/s 2] [m/s 2] [m/s 2] [m/s 2] Conventional 0 500 1000 1500 2000 −0.5 0 0.5 1 [m] PID 0 500 1000 1500 2000 −0.5 0 0.5 1 [m] PIDSH 0 500 1000 1500 2000 −0.5 0 0.5 1 [m] LQ
Fig. 12. Time history of the lateral acceleration perceived by passengers. Top left: passive vehicle 300 km/h. Top right: PID regulator, 340 km/h. Bottom left: PIDSH regulator, 340 km/h. Bottom right: LQ regulator, 340 km/h.
0 500 1000 1500 2000 −40 −30 −20 −10 0 10 20 [m] [m] [m] [m] [mrad] Conventional 0 500 1000 1500 2000 −40 −30 −20 −10 0 10 20 [mrad] Reference PI D 0 500 1000 1500 2000 −40 −30 −20 −10 0 10 20 [mrad] Reference PIDSH 0 500 1000 1500 2000 −40 −30 −20 −10 0 10 20 [mrad] Reference LQ
Fig. 11. Time history of the carbody roll rotation. Top left: passive vehicle 300 km/h. Top right: PID regulator, 340 km/h. Bottom left: PIDSH regulator, 340 km/h. Bottom right: LQ regulator, 340 km/h.
Considering the case of a vehicle running on a typical curve in a high-speed railway network, simulation results show that the active suspension permits an increase in vehicle speed from 300 km/h to 340 km/h for the same or improved ride comfort. The mean power required to operate the active suspension is in
the range of 0.5–1.0 kW per bogie, which is fully feasible for the
application envisaged here. The control strategy based on
feed-forwardþPID feed-back with sky-hook damping provides the best
performance in terms of ride comfort for an acceptable actuation power requirement. The LQ regulator is effective in mitigating carbody vibration induced by track irregularity, but its overall
performance is affected by significant actuation delay during the
negotiation of curve transitions.
Further work is planned on the integration of active tilt and active lateral control for the secondary suspension and to inves-tigate the rejection of disturbances generated by track irregularity in a wider frequency range, also considering the effects of carbody deformability.
Appendix A
In this appendix, the details of derivation of Eqs.(1) and (2)are
reported.Fig. A1shows thefluid flow rates in the active
suspen-sion hydraulic circuit.
For a compressible fluid, the bulk modulus β is defined as
(Merritt, 1967) β ¼ ρ0∂p ∂ρ ρ0 ðA:1Þ
Considering mass conservation for each actuator chamber under
the assumption of partially compressible fluid results in the
following equations: Qin1ciðp1p2Þcep1¼ A_ulþ VAAul β _p1 ðA:2Þ Qout1ciðp1p2Þcep1¼ þA_urþ VAþAur β _p1 ðA:3Þ Qin2þciðp1p2Þcep2¼ A_urþ VAAur β _p2 ðA:4Þ Qout2þciðp1p2Þcep2¼ A_ulþ VAþAul β _p2 ðA:5Þ
where VAis the volume of the actuator chambers in the reference
configuration, Qin1, Qout1, Qin2, and Qout2 are thefluid flow rates
shown inFig. A1, and all other symbols are introduced inSection 3.
Furthermore, mass conservation for the two reservoirs leads to the following two equations:
Qin1þQout1þQs1¼ VR β _p1 ðA:6Þ Qin2þQout2þQs2¼ VR β _p2 ðA:7Þ
with VRthe volume of each reservoir.
Introducing(A.6)in the sum of(A.2)and(A.3)gives Eq.(1). In
the same way, introducing(A.7)in the sum of(A.4)and(A.5)gives
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Table 2
Comparison of the comfort indexes and power required for actuation.
Speed [km/h] PCT(seated passengers) [dimensionless]
r.m.s. lateral acceleration [m/s2]
r.m.s. of power required for actuation (per bogie) [kW]
Passive 300 2.98 0.159 –
PID (low actuation level) 340 2.51 0.200 0.536
PID (high actuation level) 340 2.49 0.222 0.996
PIDSH (low actuation level) 340 2.33 0.194 0.548
PIDSH (high actuation level) 340 2.18 0.183 0.940
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